#1. A composite wall consists of three layers in series: 8 cm brick (k = 0.9 W/(m·K)), 4 cm air gap (k = 0.026 W/(m·K)), and 2 cm plaster (k = 0.5 W/(m·K)). The inside and outside convection coefficients are 10 W/(m²·K) and 25 W/(m²·K) respectively. For a wall area of 2 m² with inside air at 25°C and outside air at -5°C, calculate the heat transfer rate.
#2. A cylindrical reactor vessel has inner radius 0.8 m, outer radius 1.0 m, with k = 45 W/(m·K). It is covered with 15 cm of insulation (k = 0.08 W/(m·K)). The inner surface is at 350°C and outer convection coefficient is 12 W/(m²·K) with ambient at 30°C. For L = 3 m, calculate the heat loss.
#3. Two large parallel plates (ε₁ = 0.85, ε₂ = 0.65) at temperatures 700°C and 250°C exchange radiation. A single radiation shield (εs = 0.05 on both sides) is inserted between them. Calculate the net radiation heat flux with the shield. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#4. A composite spherical shell has inner steel layer (r₁ = 6 cm to r₂ = 8 cm, k₁ = 50 W/(m·K)) and outer insulation (r₂ = 8 cm to r₃ = 12 cm, k₂ = 0.04 W/(m·K)). Inner convection h₁ = 300 W/(m²·K) with fluid at 200°C. Outer convection h₂ = 15 W/(m²·K) with ambient at 25°C. Calculate heat transfer rate.
#5. A plane wall (L = 10 cm, k = 20 W/(m·K)) has uniform heat generation of 8 × 10⁶ W/m³. The left surface is insulated and the right surface has convection with h = 500 W/(m²·K) and T∞ = 40°C. Calculate the maximum temperature in the wall.
#6. A triangular fin of base thickness 6 mm, length 4 cm, and width 3 cm is made of aluminum (k = 200 W/(m·K)). The base temperature is 180°C and ambient is 30°C with h = 40 W/(m²·K). Using the corrected fin length approach Lc = L + t/4, calculate the approximate heat transfer rate.
#7. An annular fin (k = 180 W/(m·K)) has inner radius 2 cm, outer radius 6 cm, and thickness 3 mm. The base is at 150°C with h = 50 W/(m²·K) and T∞ = 30°C. If the fin efficiency is 72%, calculate the heat transfer from both sides of the fin.
#8. A pin fin array consists of 25 aluminum pins (k = 200 W/(m·K)), each 4 mm diameter and 25 mm long, attached to a base at 130°C. Ambient is at 30°C with h = 80 W/(m²·K). If total unfinned base area is 5 cm², calculate total heat transfer from the finned surface.
#9. For a rectangular fin with m = 25 m⁻¹ and L = 5 cm exposed to convection with h = 60 W/(m²·K) and Ac = 8 × 10⁻⁵ m², compare heat transfer for adiabatic tip versus infinite fin assumptions. What is the ratio Q_adiabatic/Q_infinite?
#10. An array of 15 cylindrical pin fins (D = 3 mm, L = 20 mm, k = 120 W/(m·K)) with fin efficiency 82% is attached to a surface. The unfinned area between fins is 8 cm². If h = 100 W/(m²·K), base temperature is 180°C, and ambient is 35°C, calculate overall surface efficiency.
#11. A counter-flow heat exchanger heats oil (ṁ = 2 kg/s, cp = 2100 J/(kg·K)) from 30°C to 80°C using hot water (ṁ = 1.5 kg/s, cp = 4180 J/(kg·K)) entering at 120°C. The overall heat transfer coefficient is 400 W/(m²·K). Calculate the required heat transfer area using LMTD method.
#12. A shell-and-tube heat exchanger (1 shell, 2 tube passes) has hot fluid entering at 200°C and exiting at 90°C, while cold fluid enters at 35°C and exits at 85°C. The LMTD correction factor F = 0.85. If U = 450 W/(m²·K) and A = 15 m², calculate the heat transfer rate.
#13. A cross-flow heat exchanger (both fluids unmixed) has NTU = 2.5 and Cr = 0.6. Using ε = 1 – exp[(1/Cr) × NTU⁰·²² × {exp(-Cr × NTU⁰·⁷⁸) – 1}], estimate the effectiveness.
#14. A counter-flow regenerative heat exchanger has effectiveness ε = 0.90 and Cr = 0.8. Hot gas (ṁ = 3 kg/s, cp = 1050 J/(kg·K)) enters at 500°C and cold air (ṁ = 2.5 kg/s, cp = 1008 J/(kg·K)) enters at 30°C. Calculate the outlet temperature of both streams.
#15. A heat exchanger must heat 5 kg/s of cold water from 20°C to 65°C using condensing steam at 110°C. If U = 2500 W/(m²·K), calculate the required area assuming constant steam temperature (Cr = 0).
#16. A large steel plate (k = 45 W/(m·K), α = 1.2 × 10⁻⁵ m²/s) of thickness 10 cm is initially at 400°C. It is suddenly exposed to convection on both sides with h = 500 W/(m²·K) and T∞ = 30°C. Using Heisler charts with Bi = 0.556 and Fo = 0.48 gives θ* = 0.6 at center. Calculate center temperature.
#17. A long cylinder (r = 3 cm, k = 20 W/(m·K), α = 8 × 10⁻⁶ m²/s) initially at 300°C is quenched in oil at 40°C with h = 800 W/(m²·K). Calculate Bi and Fo after 90 seconds, and determine if center temperature can be found using one-term approximation.
#18. A sphere (D = 8 cm, k = 50 W/(m·K), α = 1.5 × 10⁻⁵ m²/s) initially at 500°C is cooled in fluid at 50°C. After 200 seconds with h = 250 W/(m²·K), Heisler charts give θ*₀ = 0.35 at center and position correction θ*(r=R)/θ*(r=0) = 0.70. Calculate surface temperature.
#19. A semi-infinite solid (α = 1 × 10⁻⁵ m²/s, k = 40 W/(m·K)) initially at 25°C is exposed to convection with h = 500 W/(m²·K) and T∞ = 200°C. Calculate temperature at 2 cm depth after 100 seconds. Given: erfc(0.316) = 0.654, exp(0.8) = 2.23.
#20. A wall initially at 20°C has its left surface suddenly raised to 300°C. The wall thickness is 20 cm with α = 5 × 10⁻⁶ m²/s. After 500 seconds, using superposition with Fo = 0.0625 and x/L = 0.5, find the temperature at the midplane if the right surface remains at 20°C.
#21. A cylindrical furnace (D = 1.5 m, L = 2.5 m) has walls at 1200 K with ε = 0.8. The two circular ends are at 600 K with ε = 0.6. Using F_end-wall = (1 – F_end-end) and approximate F_end-end = 0.17 for L/D = 1.67, calculate net radiation from the side wall. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#22. Three surfaces form an enclosure: A₁ = 2 m² at T₁ = 500 K with ε₁ = 0.9, A₂ = 3 m² at T₂ = 350 K with ε₂ = 0.7, and A₃ is a reradiating surface. F₁₂ = 0.4, F₁₃ = 0.6. Calculate net heat transfer from surface 1. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#23. A small gray sphere (A = 0.05 m², ε = 0.8, T₁ = 800 K) is located at the center of a large spherical enclosure (T₂ = 400 K). A thin concentric spherical shield (εs = 0.05) is placed midway. Calculate the net radiation with the shield. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#24. Two parallel disks (A₁ = A₂ = π × 0.25² = 0.196 m²) are separated by distance 0.5 m. Using the coaxial disk formula, F₁₂ = 0.17. Disk 1 is at 600 K with ε₁ = 0.9, disk 2 is at 400 K with ε₂ = 0.85. Calculate net radiation between them. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#25. A gas turbine blade with surface area 0.002 m² at 1100 K and ε = 0.75 is cooled by internal convection (h = 1500 W/(m²·K), coolant at 600 K). The blade also loses heat by radiation to surroundings at 500 K. Calculate total heat loss per blade. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#26. A vertical plate 0.6 m tall is maintained at 90°C in ambient air at 20°C. For air: β = 0.0029 K⁻¹, ν = 1.8 × 10⁻⁵ m²/s, k = 0.028 W/(m·K), Pr = 0.71. Using Ra = GrPr and Nu = 0.59Ra⁰·²⁵ for laminar flow, calculate the average heat transfer coefficient.
#27. Air flows at 8 m/s over a 0.5 m long vertical heated plate at 100°C in ambient at 30°C. Calculate Gr/Re² to determine the dominant mode. For air: ν = 2 × 10⁻⁵ m²/s, β = 0.003 K⁻¹, g = 9.81 m/s².
#28. A horizontal cylinder (D = 5 cm) at 140°C is exposed to ambient air at 30°C. Using the correlation Nu = 0.53Ra⁰·²⁵ for Ra in range 10⁴-10⁹, with Ra = 8 × 10⁶, calculate the heat transfer coefficient. k = 0.029 W/(m·K).
#29. Natural convection occurs in a horizontal enclosure (H = 3 cm) heated from below with ΔT = 40°C. For air: α = 2.5 × 10⁻⁵ m²/s, ν = 1.8 × 10⁻⁵ m²/s, β = 0.0032 K⁻¹. Calculate the Rayleigh number and predict if convection cells will form (Ra_crit = 1708).
#30. For assisting mixed convection (buoyancy aids flow) over a vertical plate, the combined Nusselt number is approximated by Nu³ = Nu_forced³ + Nu_natural³. If Nu_forced = 150 and Nu_natural = 80, calculate the combined Nu.
#31. Water boils at 1 atm on a horizontal copper surface at 118°C. Using the Rohsenow correlation for nucleate boiling with Csf = 0.013 and n = 1.0, the heat flux is proportional to (ΔTe)³/Csf². If the excess temperature increases from 18°C to 25°C, by what factor does the heat flux increase?
#32. Saturated steam at 1 atm condenses on a 0.4 m tall vertical plate maintained at 85°C. Using Nusselt analysis with water properties: ρ = 958 kg/m³, k = 0.68 W/(m·K), μ = 2.8 × 10⁻⁴ Pa·s, hfg = 2.26 × 10⁶ J/kg, calculate average h.
#33. For film boiling of water on a horizontal cylinder (D = 2 cm) at 300°C surface temperature (ΔTe = 200°C), the Bromley correlation gives h_fb = 0.62[k³ρg(ρl-ρv)hfg’/(μDΔTe)]⁰·²⁵. If h_fb = 180 W/(m²·K) and radiation contributes h_rad = 50 W/(m²·K), estimate effective h using h_eff = h_fb + 0.75h_rad.
#34. In a horizontal tube condenser, 2000 kg/h of saturated steam at 50 kPa is condensed. If hfg = 2.305 MJ/kg and U = 3500 W/(m²·K), with cooling water temperature rise limiting LMTD to 15°C, calculate required tube surface area.
#35. Dropwise condensation on a vertical surface provides heat transfer coefficient 10 times higher than film condensation. If film condensation h = 8000 W/(m²·K), and the surface has 70% dropwise and 30% film coverage, calculate effective h.
#36. For turbulent flow in a tube, the Colburn analogy states St × Pr^(2/3) = f/8, where St = Nu/(Re × Pr). If f = 0.02, Re = 50,000, and Pr = 5, calculate the Nusselt number.
#37. Using the Chilton-Colburn analogy for heat and mass transfer, if Sherwood number Sh = 180 for mass transfer with Sc = 600, estimate the Nusselt number for heat transfer with Pr = 0.7 for the same flow conditions.
#38. For laminar boundary layer flow over a flat plate with Pr = 0.01 (liquid metal), the thermal boundary layer thickness ratio δt/δ is approximately Pr⁻¹/². Calculate this ratio and comment on implications.
#39. A fluid with Pr = 100 (heavy oil) flows over a flat plate. Using δt/δ ≈ Pr⁻¹/³, calculate the thermal boundary layer to velocity boundary layer thickness ratio.
#40. For flow through a tube, the Graetz number Gz = Re × Pr × (D/L) determines thermal development. If Re = 1500, Pr = 6, D = 2 cm, and L = 1 m, calculate Gz and determine if flow is thermally developing.
#41. A rectangular parallelepiped building foundation (20 m × 15 m × 3 m deep) transfers heat to the ground surface. Using the shape factor S = 0.078L(A)⁰·⁵ for slab-on-grade with L = perimeter, calculate the shape factor.
#42. An isothermal pipe (D = 15 cm, L = 10 m, Ts = 80°C) is buried with centerline 1.2 m below ground surface (Tg = 15°C). If soil k = 1.5 W/(m·K), using S = 2πL/ln(4z/D), calculate heat loss rate.
#43. Two parallel isothermal cylinders (D₁ = D₂ = 10 cm) with centers 50 cm apart and L = 5 m exchange heat. Using S = 2πL/cosh⁻¹[(D²-d₁²-d₂²)/(2d₁d₂)] where D = center distance, calculate shape factor for one cylinder.
#44. A hemisphere of radius 30 cm is buried in soil (k = 2 W/(m·K)) with flat surface at ground level. Using S = 2πr for this configuration, if the hemisphere surface is at 150°C and ground surface at 10°C, calculate heat loss.
#45. Heat flows through a corner joint of two walls of equal thickness t = 15 cm. Using S = 0.15t for inside corner and S = 0.54t for outside corner, calculate the ratio of heat loss through outside corner to inside corner per unit length.
#46. A plane wall (L = 8 cm) has temperature-dependent thermal conductivity k = k₀(1 + βT) where k₀ = 15 W/(m·K) and β = 0.003 K⁻¹. Surface temperatures are 350°C and 80°C. Calculate heat flux using mean k at average temperature.
#47. For a cylindrical rod with k = k₀(1 – αT) where k₀ = 30 W/(m·K) and α = 0.001 K⁻¹, inner surface (r = 2 cm) at 400°C and outer surface (r = 5 cm) at 100°C. Using km at Tm = 250°C, calculate heat transfer per unit length.
#48. A sphere (r₁ = 5 cm, r₂ = 10 cm) has k = 20(1 + 0.002T) W/(m·K). Inner surface at 500°C, outer at 100°C. Calculate heat transfer rate using average k.
#49. Heat generation in a plane wall varies as q”’ = q₀(1 + βT) where q₀ = 5 × 10⁶ W/m³ and β = 0.002 K⁻¹. Wall thickness 4 cm, k = 20 W/(m·K), both surfaces at 100°C. Estimate maximum temperature.
#50. A wire (D = 2 mm) carries current generating heat at q”’ = 10⁸ W/m³. Wire k varies as k = 15 + 0.02T W/(m·K). Surface temperature is 150°C. Estimate centerline temperature using an iterative approach or graphical solution.
#51. Two aluminum blocks are pressed together with a contact pressure of 10 MPa. The contact conductance is hc = 40,000 W/(m²·K) at this pressure. If each block is 5 cm thick with k = 200 W/(m·K) and the outer surfaces are at 300°C and 50°C, calculate the temperature drop across the interface.
#52. In a CPU heat sink assembly, thermal paste (k = 4 W/(m·K), thickness 0.1 mm) fills gaps between surfaces. If contact resistance without paste is 0.5 × 10⁻⁴ m²·K/W and paste reduces effective air gaps, calculate total interface resistance with paste.
#53. A bolted joint connects two steel plates (each 2 cm thick, k = 50 W/(m·K)). Under the bolt head (A = 1 cm²), contact conductance is 25,000 W/(m²·K). Away from bolts (A = 9 cm²), conductance is 3,000 W/(m²·K). Calculate effective overall contact conductance.
#54. An electronic package (A = 25 cm², P = 20 W) is mounted on a heat sink through a thermal interface with conductance 15,000 W/(m²·K). If heat sink surface is at 45°C, calculate chip temperature at the interface.
#55. Thermal contact resistance between rough surfaces decreases with increasing pressure approximately as Rc ∝ P⁻⁰·⁹⁵. If Rc = 2 × 10⁻⁴ m²·K/W at 1 MPa, calculate Rc at 5 MPa.
#56. A heat exchanger has initial U = 1500 W/(m²·K). After 6 months, fouling on water side adds 0.0002 m²·K/W and oil side adds 0.0003 m²·K/W. Calculate the fouled U and percentage decrease in heat transfer capacity.
#57. A condenser tube (D = 25 mm, k = 110 W/(m·K)) has thickness 2 mm. Steam-side h = 8000 W/(m²·K), water-side h = 5000 W/(m²·K). Design fouling factors: 0.0001 m²·K/W (steam), 0.0002 m²·K/W (water). Calculate design U.
#58. A heat exchanger designed for Q = 500 kW with U = 800 W/(m²·K) and LMTD = 40°C operates with fouled U = 600 W/(m²·K). If inlet conditions remain same, calculate the actual heat transfer rate.
#59. Scaling in a boiler tube reduces effective diameter from 50 mm to 46 mm. If water-side h ∝ D⁻⁰·², by what factor does the water-side convection resistance increase?
#60. A heat exchanger with excess surface (A_actual = 1.2 × A_required) operates with cleanliness factor CF = 0.85 when fouled. Calculate the actual heat transfer as a percentage of design capacity.
#61. A thin metal plate (ε = 0.7) is exposed to solar radiation of 800 W/m² on one side and loses heat by convection (h = 15 W/(m²·K)) and radiation to sky (Tsky = 250 K) on the other side. Ambient is 30°C. Calculate equilibrium temperature of the plate. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#62. A horizontal pipe (D = 10 cm, L = 3 m, ε = 0.9) at 200°C loses heat to surroundings at 25°C by combined natural convection (h = 8 W/(m²·K)) and radiation. Calculate total heat loss rate. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#63. A cryogenic pipe (outer D = 8 cm, Ts = -150°C) is surrounded by radiation shield (D = 12 cm, ε = 0.05) in vacuum (no convection). Outer surface at 25°C with ε = 0.9. Calculate heat gain per meter. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#64. A vertical heated surface (H = 1 m, Ts = 120°C, ε = 0.85) loses heat to ambient at 20°C. Natural convection h = 6 W/(m²·K). If radiation shields (ε = 0.1) are added, reducing radiation to 10% of original, calculate percentage reduction in total heat loss.
#65. An electronic component (A = 5 cm², P = 8 W) is cooled by combined natural convection (h = 12 W/(m²·K)) and radiation (ε = 0.8) to ambient at 25°C. Estimate component temperature. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#66. For a rectangular fin to achieve maximum heat transfer per unit mass, the optimal profile satisfies t ∝ (1-x/L)². For such an optimized fin with base t₀ = 6 mm, L = 5 cm, W = 3 cm, k = 180 W/(m·K), h = 50 W/(m²·K), θb = 120°C, estimate heat transfer.
#67. A spine fin (triangular profile) has base diameter 8 mm, length 4 cm, k = 160 W/(m·K), h = 40 W/(m²·K). Base at 150°C, ambient 25°C. Using corrected length Lc = L + D/4 and modified m, calculate approximate heat transfer.
#68. For maximum heat dissipation from a given surface area, the optimal fin spacing S for natural convection is S_opt ≈ 2.7(Ra_L)⁻⁰·²⁵L where L is fin length. For L = 5 cm, Ra_L = 10⁵, calculate optimal spacing.
#69. A heat sink with parallel rectangular fins has fin length 25 mm, spacing 5 mm, fin thickness 1 mm, and k = 200 W/(m·K). If h = 30 W/(m²·K) between fins and h = 50 W/(m²·K) at fin tips, calculate fin efficiency.
#70. For a cylindrical pin fin, the volume for given heat dissipation is minimized when mL ≈ 1.42. For k = 200 W/(m·K) and h = 100 W/(m²·K), if L = 20 mm, calculate the optimal pin diameter.
#71. In finite difference solution of 2D steady conduction with Δx = Δy, the equation at an interior node relates the node temperature to surrounding nodes as T₀ = (T₁ + T₂ + T₃ + T₄)/4. For nodes at 100°C, 150°C, 80°C, and 120°C around a central node, calculate T₀.
#72. A 1D transient conduction problem is solved using explicit finite difference method. The stability criterion requires Fo ≤ 1/2 where Fo = αΔt/Δx². If α = 1 × 10⁻⁵ m²/s and Δx = 1 cm, calculate maximum allowable time step.
#73. In Gauss-Seidel iteration for solving heat conduction, if initial guess for interior nodes is 100°C and boundary nodes are 200°C (left) and 50°C (right) for a 5-node (3 interior) system, calculate the first-iteration value of the middle node.
#74. For implicit (Crank-Nicolson) solution of 1D transient conduction, the method averages explicit and implicit formulations. If explicit gives T = 78°C and implicit gives T = 82°C at a node after one time step, what is the Crank-Nicolson result?
#75. A convection boundary condition is approximated in finite differences as k(T₁-T₀)/Δx = h(T₀-T∞). For k = 50 W/(m·K), h = 100 W/(m²·K), Δx = 5 mm, T₁ = 200°C, T∞ = 25°C, calculate T₀.
#76. A phase change material (PCM) storage unit uses paraffin (hfg = 200 kJ/kg, ρ = 800 kg/m³, melting point 35°C). A cylindrical container (D = 20 cm, H = 30 cm) is initially solid at 25°C. How much energy is stored when fully melted and heated to 45°C? (liquid cp = 2000 J/(kg·K), solid cp = 1500 J/(kg·K))
#77. A concrete wall (ρ = 2400 kg/m³, c = 880 J/(kg·K), k = 1.4 W/(m·K), thickness 20 cm) is used for thermal storage. Calculate the thermal time constant τ = ρcL²/k for this wall.
#78. Ice at 0°C is used to cool air in a thermal storage system. If 100 kg of ice (hfg = 334 kJ/kg) is completely melted and the resulting water warms to 10°C (cp = 4.18 kJ/(kg·K)), calculate total cooling capacity.
#79. A packed bed thermal storage (stones, void fraction 0.4) has bulk density ρb = 1440 kg/m³ and cp = 920 J/(kg·K). For a bed volume of 5 m³ with ΔT = 80°C, calculate thermal storage capacity.
#80. A stratified water tank (H = 2 m, D = 1 m) for solar heating has top at 65°C and bottom at 35°C (linear stratification). Using cp = 4180 J/(kg·K), calculate available thermal energy above 40°C reference.
#81. A thermoelectric cooler (Peltier device) with ZT = 0.8 operates between Tc = 0°C (cold side) and Th = 50°C (hot side). The coefficient of performance for cooling is approximately COP = Tc/(Th-Tc) × [√(1+ZT) – Th/Tc]/[√(1+ZT) + 1]. Calculate COP.
#82. A heat pipe has evaporator length 10 cm (h = 5000 W/(m²·K)), condenser length 15 cm (h = 2000 W/(m²·K)), and adiabatic section 20 cm. Pipe OD = 8 mm. If evaporator is at 100°C and condenser coolant at 30°C, calculate heat transfer rate.
#83. In a regenerative heat exchanger with periodic flow reversal, thermal effectiveness depends on NTU and Cr (for balanced flow, Cr = 1). For a rotary regenerator with NTU = 4 and Cr = 1, ε ≈ NTU/(1+NTU). Calculate effectiveness.
#84. A micro-channel heat sink has 50 rectangular channels (W = 100 μm, H = 300 μm, L = 10 mm). Water flows at Re = 500 (laminar, Nu = 4.5). If k_water = 0.6 W/(m·K), calculate average h in the channels.
#85. A thermocouple junction (D = 1 mm, ρ = 8900 kg/m³, c = 385 J/(kg·K), k = 22 W/(m·K)) is exposed to gas at 500°C with h = 500 W/(m²·K). Junction initially at 25°C. Calculate time constant and temperature after 0.5 s.
#86. A thermocouple measures temperature in a gas stream at 400°C with h = 150 W/(m²·K). The thermocouple also radiates to duct walls at 300°C with ε = 0.7. Calculate the measurement error due to radiation. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#87. An unshielded temperature sensor in a hot gas stream shows 450°C. Adding a radiation shield increases reading to 475°C. If true gas temperature is 500°C, calculate the ratio of shield effectiveness to unshielded sensor error.
#88. Heat flux measurement using a thin-foil gauge requires correction for losses. If gauge reads 5000 W/m², convection from rear is 200 W/m², radiation from rear is 150 W/m², and lateral losses are 100 W/m², calculate actual heat flux.
#89. A guarded hot plate apparatus measures thermal conductivity. Sample thickness 25 mm, area 0.04 m², power 50 W, temperatures 85°C and 25°C. Edge guard eliminates lateral losses. Calculate thermal conductivity.
#90. Uncertainty in heat flux q = k(ΔT/L) propagates from uncertainties in k (±3%), ΔT (±2%), and L (±1%). Calculate combined uncertainty in q.
#91. A steel slab (thickness 20 cm, k = 45 W/(m·K)) exits a reheat furnace at 1200°C average temperature. It is cooled by water sprays providing h = 2000 W/(m²·K) with water at 25°C. Using Bi = 4.44 and Fo = 0.5, estimate center temperature after 200 s if α = 1.2 × 10⁻⁵ m²/s.
#92. An ASME Section VIII pressure vessel is designed for 350°C with maximum ΔT = 50°C through the wall during normal operation. If wall thickness is 60 mm and k = 40 W/(m·K), calculate maximum allowable heat flux.
#93. A gas turbine blade receives 80 kW/m² heat flux from combustion gases. Internal cooling passages maintain h = 5000 W/(m²·K) with coolant at 600°C. Blade wall is 3 mm with k = 25 W/(m·K). Calculate outer surface temperature.
#94. A nuclear fuel rod (D = 10 mm, k = 3 W/(m·K)) generates 5 × 10⁸ W/m³. Cladding (0.5 mm, k = 15 W/(m·K)) has coolant h = 30,000 W/(m²·K) at 300°C outside. Calculate fuel centerline temperature.
#95. An automotive radiator rejects 40 kW with airflow providing h_air = 80 W/(m²·K) and coolant h_water = 4000 W/(m²·K). Thin fins double the air-side area. If LMTD = 35°C, calculate required bare tube area.
#96. A double-pipe heat exchanger is converted from parallel-flow to counter-flow by reversing one stream. Original parallel ε = 0.55 with NTU = 1.5 and Cr = 0.7. Calculate new counter-flow effectiveness.
#97. A cylindrical furnace wall has inner refractory (r₁ = 0.5m to r₂ = 0.7m, k₁ = 1.5 W/(m·K)), insulating brick (r₂ to r₃ = 0.85m, k₂ = 0.3 W/(m·K)), and steel shell (r₃ to r₄ = 0.86m, k₃ = 45 W/(m·K)). Inner at 1200°C, outer h = 10 W/(m²·K), T∞ = 30°C. Calculate heat loss per meter of height.
#98. An array of 100 LEDs (each 0.5 W) mounted on an aluminum heat sink (L = 15 cm × W = 10 cm × H = 2 cm) must not exceed 75°C in 40°C ambient. Natural convection provides h = 10 W/(m²·K). Calculate the required fin surface area if fins have η = 0.8.
#99. In a solar collector, absorber plate (α = 0.95, ε = 0.1) at 80°C receives 900 W/m² solar radiation. Convection loss to ambient (35°C) has h = 5 W/(m²·K). Calculate net heat gain per unit area. σ = 5.67 × 10⁻⁸ W/(m²·K⁴).
#100. A precision casting mold (k = 30 W/(m·K), α = 8 × 10⁻⁶ m²/s) at 25°C suddenly contacts molten metal at 700°C. Interface contact conductance h = 5000 W/(m²·K). Calculate mold surface temperature after 10 seconds of contact.
